Seven dimensional considerations of Einstein's connection I The recurrence relations of the first kind in n g UFT

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SEVEN-DIMENSIONAL CONSIDERATIONS OF EINSTEIN’S

CONNECTION. I. THE RECURRENCE RELATIONS

OF THE FIRST KIND IN

n

-

g

-UFT

MI AE KIM, KEUM SOOK SO, CHUNG HYUN CHO, and KYUNG TAE CHUNG

Received 12 April 2001 and in revised form 27 September 2001

Einstein’s connection in a generalized Riemannian manifoldXnhas been investi-gated by many authors for lower-dimensional casesn=2,...,6. In a series of pa-pers, we obtain a surveyable tensorial representation of 7-dimensional Einstein’s connection in terms of unified field tensor, with main emphasis on the derivation of powerful and useful recurrence relations which hold inX7. In this paper, we

give a brief survey of Einstein’s unified field theory and derive the recurrence rela-tions of the first kind which hold in a generalXn. All considerations in this paper are restricted to the first and second classes only since the case of the third class, the simplest case, was already studied by many authors.

2000 Mathematics Subject Classification: 83E50, 83C05, 58A05.

1. Introduction. In Appendix II to his last book, Einstein [16] proposed a new unified field theory that would include both gravitation and electromag-netism. Although the intent of this theory is physical, its exposition is mainly geometrical. Characterizing Einstein’s unified field theory as a set of geomet-rical postulates in the space timeX4, Hlavatý [17] gave its mathematical

foun-dation for the first time. Since then Hlavatý and number of mathematicians contributed for the development of this theory and obtained many geometri-cal consequences of these postulates.

GeneralizingX4ton-dimensional generalized Riemannian manifoldXn,

generalizations, it has not been possible yet to represent the general n-dimen-sional Einstein’s connection in a surveyable tensorial form in terms of the unified field tensorgλµ. This is probably due to the complexity of the higher

dimensions.

However, the lower-dimensional cases of the Einstein’s connection in n-g-UFT were investigated by many authors, 2-dimensional case by Jakubowicz [18] and Chung et al. [4], 3-dimensional case by Chung et al. [9, 10,11], and 4-dimensional case by Hlavatý [17] and many other geometricians. Recently, Chung et al. also studied the Einstein’s connection in 4-∗g-UFT [1], in 3-and 5-∗g-UFT [8], and in 6-g-UFT [14,15], and obtained respective Einstein’s con-nection in a surveyable tensorial form.

The purpose of the present paper, the first part of a series of three papers, is to introduce a brief survey of Einstein’sn-g-UFT and derive several recurrence relations of the first kind which hold inn-g-UFT. In the second and third parts, we derive powerful and useful recurrence relations of the second and third kind which hold in 7-g-UFT, prove a necessary and sufficient condition for the existence and uniqueness of the 7-dimensional Einstein’s connection, establish a linear system of 66 equations for the solution of the 7-dimensional Einstein’s connection employing the powerful recurrence relations, and finally solve the system. We hope that the results obtained in this series of papers, especially the 7-dimensional recurrence relations of the second and third kind in part II, will contribute to the study of generalizing the corresponding results in a generaln-dimensional case whennis odd.

All considerations in this and subsequent papers are dealt with for the first and second classes only.

2. Preliminaries. This section is a brief collection of basic concepts, nota-tions, and results, which are needed in our subsequent considerations. They are due to Chung [1,8], Hlavatý [17], and Mishra [19]. All considerations in this section are dealt with for a generaln >1.

2.1.n-dimensional g-unified field theory. The Einstein’s n-dimensional unified field theory, denoted byn-g-UFT, is ann-dimensional generalization of the usual Einstein’s 4-dimensional unified field theory in the space-timeX4.

It is based on the following three principles as indicated by Hlavatý [17].

Principle2.1. LetX_nbe ann-dimensional generalized Riemannian

mani-fold referred to a real coordinate systemxν_{, which obeys the coordinate}

trans-formation xν _→xν

(throughout the present paper, Greek indices are used for the holonomic components of tensors, while Roman indices are used for the nonholonomic components of a tensor inXn. All indices take the values

1,2,...,nand follow the summation convention with the exception of nonholo-nomic indicesx,y,z,t) for which

det

_∂x ∂x

Inn-g-UFT, the manifoldXnis endowed with a real nonsymmetric tensorgλµ,

calledthe unified field tensor ofXn. This tensor may be decomposed into its

symmetric parthλµand skew-symmetric partkλµ,

gλµ=hλµ+kλµ. (2.2)

For the case ofn=4, the gravitational field is described by 10 components of a symmetric quadratic tensor, while the electromagnetic field is defined by a quadratic skew-symmetric tensor described by 6 components. Therefore, we may expect that a unified theory comprising gravitation and electromag-netism to be based on 16 field quantities. Several attempts in this direction were made by Einstein and ultimately discarded. As given in (2.2), his last at-tempt, which he regarded as final, was to introduce the unified field by a real nonsymmetric tensorgλµ, the gravitational field by its symmetric parthλµ, and

the electromagnetic field by a skew-symmetric quadratic tensorkλµwhich may

be represented as a linear combination of its skew-symmetric partkλµ, where

g=detgλµ≠0, h=dethλµ≠0, k=detkλµ. (2.3)

We may define a unique tensorhλν_=hνλ_by

hλµhλν=δνµ. (2.4)

Inn-g-UFT, the tensorshλµandhλν will serve for raising and/or lowering

indices of tensors inXnin the usual manner.

Principle_2.2. _{The differential geometric structure on}X_{n is imposed by}

the tensorgλµby means of a connectionΓλµν , defined by a system of equations

Dωgλµ=2Sωµαgλα. (2.5)

For 4-dimensional case, the tensor fieldgλµ imposes a differential

geomet-ric structure on space-timeX4 through the Einstein’s connectionΓλµν, which

Einstein originally defined by the following system of 64 equations, called Ein-stein’s equations:

∂ωgλµ−Γλωα gαµ−Γωµα gλα=0. (2.6)

The equivalence of (2.5) and (2.6) was proved by Hlavatý [17, page 50]. This system of 64 equations is similar in structure to the system of 40 equations which defines the Christoffel symbols in the gravitational theory. The only striking difference between these two systems is the position of the indices ωµ in Γν

ωµ. However, this peculiar order of the indicesωµ has a profound

Meaning of Relativity), Princeton University Press, 1953, as follows: “this fact is the mathematical expression of the indifference of the theory to the sign of electricity.” This is an intuition of genius, since Einstein was not in possession of the tensorkλµof Electromagnetism.

HereDωdenotes the symbolic vector of the covariant derivative with respect

toΓν

λµ, andSλµν=Γ[λµ]ν is the torsion tensor ofΓλµν. The connectionΓλµν satisfying

(2.5) is calledthe Einstein’s connection. Under certain conditions the system (2.5) admits a unique solutionΓν

λµ.

Principle_2.3. _{In order to obtain}g_{λµinvolved in the solution for}Γ_λµν _certain

conditions are imposed. These conditions may be condensed to

Sλ=Sλαα=0, R[µλ]=∂[µXλ], (2.7)

whereXλis an arbitrary nonzero vector, andRωµλνandRµλare the curvature

tensors ofΓν

λµdefined by

Rωµλν=2∂[µΓ|λ|νω]+Γαν[µΓ|λ|αω], Rµλ=Rαµλα. (2.8)

2.2. Algebraic preliminaries. In this subsection, we introduce notations, concepts, and several algebraic results inn-g-UFT.

2.2.1. Notations. The following scalars, tensors, and notations are fre-quently used in our further considerations:

g=g_h, k=k_h,

Kp=k[α1α1kα2α2···kαp]αp, (p=0,1,2,...),

(0)_k

λν=δνλ, (1)kλν=kλν, (p)kλν=(p−1)kλαkαν, (p=1,2,...),

Kωµν= ∇νkωµ+∇ωkνµ+∇µkων,

σ=   

0, ifnis even, 1, ifnis odd,

(2.9)

where∇ω is the symbolic vector of covariant derivative with respect to the

Christoffel symbolsλµν defined byhλµ. It has been shown that the scalars

and tensors introduced in (2.9) satisfy

K0=1, Kn=k ifnis even, Kp=0 ifpis odd,

g=1+K2+···+Kn−σ, (p)_k

λµ=(−1)p (p)kµλ, (p)kλν=(−1)p(p)kνλ.

Denoting an arbitrary tensorTωµλ, skew-symmetric in the first two indices,

byT, we also use the following useful abbreviations:

pqr

T =pqrT ωµλ=Tαβγ(p)kωα(q)kµβ(r )kλγ,

T=Tωµλ= 000

T ,

2pqrT ω[λµ]= pqr

T ωλµ− pqr

T ωµλ,

2(pq)rT ωλµ= pqr

T ωλµ+ qpr

T ωλµ, etc.

(2.11)

We then have

pqr

T ωλµ= − qpr

T λωµ. (2.12)

2.2.2. Classification, basic vectors, and basic scalars

Definition2.4. The tensorg_λµ(ork_λµ) is said to be

(1) of the first class, ifKn−σ≠0,

(2) of the second class with thejth category(j≥1), if

K2j≠0, K2j+2=K2j+4= ··· =Kn−σ=0, (2.13)

(3) of the third class, ifK2=K4= ··· =Kn−σ=0.

The solution of the system of (2.5) is most conveniently brought about in a nonholonomic frame of reference, which may be introduced by the projectivity

MAν_=k

µνAµ, (Ma scalar). (2.14)

Definition2.5. An eigenvectorAνofk_λµsatisfying (2.14) is called abasic

vectorinXn, and the corresponding eigenvalueMis termed abasic scalar.

It has been shown that the basic scalarsMare solutions of the characteristic equation

Mσ_Mn−σ_+K

2Mn−2−σ+···+Kn−2−σM2+Kn−σ=0. (2.15)

2.2.3. Nonholonomic frame of reference. In the first and second classes, we have a set ofnlinearly independent basic vectorsA

i

ν _{(i=1,...,n)and a}

unique reciprocal setAiλ(i=1,...,n), satisfying

j

AλA i

λ_=δj i,

i

AλA i

ν_=δν

λ. (2.16)

Definition_2.6. _If T_λν···

··· are holonomic components of a tensor, then its nonholonomic componentsTi···

j··· are defined by

Ti··· j···=Tλν······

i

Aν···A j

λ_{···. (2.17)}

An easy inspection shows that

Tν···

λ··· =Tji······A_iν··· j

Aλ···. (2.18)

Furthermore, ifM

x is the basic scalar corresponding toAx

ν_{, then the}

nonholo-nomic components of(p)kλνare given by

(p)_k

xi=M_xpδix, (p)kxi=M_xphxi, (p)kxi=M_xphxi. (2.19)

Without loss of generality we may choose the nonholonomic components of hλµas

h12=h34= ··· =hn−1−σ ,n−σ=1, (2.20)

σ hni0=δ

1

σ, the remaininghij=0, (2.21)

where the indexi0is taken so that Det(hij)≠0 whennis odd.

2.3. Differential geometric preliminaries. In this subsection, we present several useful results involving Einstein’s connection. These results are needed in our subsequent considerations for the solution of (2.5).

If the system (2.5) admits a solutionΓν

λµ, it must be of the form

Γν λµ=

_ν

λµ

+Sλµν+Uνλµ, (2.22)

where

Uνλµ=2 001

S ν(λµ). (2.23)

The above two relations show thatour problem of determiningΓν

λµin terms of

gλµ is reduced to that of studying the tensorSλµν. On the other hand, it has

been shown that the tensorSλµνsatisfies

S=B−3(110)S , (2.24)

where

Therefore, the Einstein’s connectionΓν

λµsatisfying (2.5) may be determined if

the solutionSλµνof the system (2.24) is found.The main purpose of the present

paper is to find a device to solve the system (2.24) whenn=7.

Furthermore, for the first two classes, the nonholonomic solution of (2.24) is given by

M

xyzSxyz=Bxyz (2.26a)

or equivalently

2M

xyzSxyz=Kxyz+3K[xyz]MxMy, (2.26b)

where

M

xyz=1+MxMy+MyMz+MzMx. (2.27)

Therefore, in virtue of (2.26), we see thata necessary and sufficient nonholo-nomic condition for the system (2.5) to have a unique solution in the first two classes is

M

xyz≠0 ∀x,y,z. (2.28)

3. The recurrence relations of the first kind inn-g-UFT. This section is devoted to the derivation of the recurrence relations of the first kind and two other useful relations which hold inn-g-UFT.All considerations in this section are also dealt with for a generaln >1.

The recurrence relations of the first kind inn-g-UFT are those which are satisfied by the tensors(p)kλν. These relations will be proved in the following

theorem.

Theorem3.1(the recurrence relations of the first kind inn-g-UFT). The

tensors(p)kλνsatisfy the following recurrence relations:

For the first class:

(n+p)_k

λν+K2(n+p−2)kλν+···+Kn−σ−2(p+σ+2)kλν+Kn−σ(p+σ )kλν=0 (3.1a)

which may be condensed to

n−σ

f=0

Kf(n+p−f )kλν=0, (p=0,1,2,...). (3.1b)

For the second class with thejth category:

(2j+p)_k

which may be condensed to

2j

f=0

Kf(2j+p−f )kλν=0, (p=1,2,...). (3.2b)

Proof

The case of the first class. _LetM

x be a basic scalar. Then, in virtue of

(2.15), we have

n−σ

f=0

KfM_xn−f=0. (3.3)

Multiplyingδixto both sides of (3.3) and making use of (2.19), we have

n−σ

f=0

Kf(n−f )kxi=0 (3.4)

whose holonomic form is

n−σ

f=0

Kf(n−f )kλα=0. (3.5)

The relation (3.1) immediately follows by multiplying(p)kαν to both sides of

(3.5).

The case of the second class with the jth category. Wheng_λµ

belongs to the second class with thejth category, the characteristic equation (2.15) is reduced to

2j

f=0

KfMn−f=Mn−2j 2j

f=0

KfM2j−f=0. (3.6)

Hence, ifM

x is a root of (3.6), it satisfies

M

x 2j

f=0

KfM_x2j−f = 2j

f=0

KfM_x2j−f+1=0. (3.7)

In virtue of (2.19), multiplication ofδi

xto both sides of (3.7) gives

2j

f=0

Kf(2j−f+1)kxi=0. (3.8)

The holonomic form of (3.8) is

2j

f=0

Consequently, the relation (3.2) follows by multiplying(p−1)kανto both sides

of (3.9).

Remark_3.2. _Wheng_{λµbelongs to the second class with the first category,}

the relation (3.2), is reduced to

(p+2)_k

λν+K2(p)kλν=0, (p=1,2,...). (3.10)

In the following two theorems we prove two useful relations.

Theorem_{3.3(for the first and second classes)}. _{In the first two classes, a}

tensorTωµν, skew-symmetric in the first two indices, satisfies

(pq)r

T ωµν=

x,y,z

TxyzM_x(pM_yq)M_zr x

Aω y

Aµ z

Aν, (3.11)

r (pq)

T ν[ωµ]=

x,y,z

Tx[yz]M_y(pM_zq)M_xr x

Aν y

Aω z

Aµ. (3.12)

Proof. Making use of (2.18) and (2.20), the relation (3.11) may be proved

as in the following way:

(pq)r

T ωµν= x,y,z (pq)r T xyz x Aω y Aµ z Aν

=1₂ x,y,z

Tijk(p)kxi(q)kyj+(q)kxi(p)kyj(r )kzk x Aω y Aµ z Aν =1 2 x,y,z Txyz M x p_M y q_+M

x q_M

y p_M

z r_Ax

ω y

Aµ z

Aν.

(3.13)

The second relation can be proved similarly.

Theorem3.4(for all classes). The tensorB_ωµν, given by (2.25), satisfies

(pq)r

B =(pq)rS +(p _q)r

S +(p _q)r

S +(pq _)r

S , (3.14)

2(pq)rB ωµν= (pq)r

K ωµν+ (p_q)r

K ωµν+

(p_q)r K ν[ωµ]+

(pq_)r

K ν[ωµ], (3.15)

where

p_{=p+1, q=q+1, r=r+1. (3.16)}

Proof. In virtue of (2.11) and (2.24), the relation (3.14) may be shown as in

the following way:

(pq)r

B =(pq)rB ωµν=1

2Bαβγ

_(p)k

ωα(q)kµβ+(q)kωα(p)kµβ(r )kνγ

=1 2

_S

αβγ+Sηγkαkβη+Sβηkαkγη+Sαηkβkγη

×(p)_k

ωα(q)kµβ+(q)kωα(p)kµβ(r )kνγ.

After a lengthy calculation, we note that the right-hand side of the above equa-tion is equal to (3.14). The relation (3.15) may be proved similarly.

Acknowledgment. _{This work was partially supported by the Basic}

Sci-ence Research Institute Program, Ministry of Education, Republic of Korea, 1997, BSRI-97-1442.

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Mi Ae Kim: Department of Mathematics, Yonsei University, Seoul 120-749, Korea

Keum Sook So: Department of Mathematics, Hallym University, Chunchon 200-702, Korea

E-mail address:[email protected]

Chung Hyun Cho: Department of Mathematics, Inha University, Inchon 402-751, Korea

E-mail address:[email protected]

Kyung Tae Chung: Department of Mathematics, Yonsei University, Seoul 120-749, Korea